In the rest of this section, we consider the pseudo-vector field \(\mathcal {A}\)  induced by the lattice relaxation of Ref. [1]}. As illustrated in Fig. REF , \(\varphi ^{\mathcal {A}} \ll \varepsilon ^{\mathcal {A}}\) , suggesting that \(\varphi ^{\mathcal {A}}\)  can be neglected, and thus \(\mathcal {A} \approx \nabla \times (\hat{z} \varepsilon ^{\mathcal {A}})\) . In addition, Table REF  shows that the Fourier series of the lattice relaxation is dominated by the lowest six \(g\) s. Therefore, we can keep only these six terms and neglect others. Furthermore, because \(\varepsilon _{\mathcal {A}}\)  is real and odd, its lowest Fourier components are purely imaginary. By \(C_3\)  symmetry, they satisfy the relation
\(& \tilde{\varepsilon }^{\mathcal {A}}_{g_1} = \tilde{\varepsilon }^{\mathcal {A}}_{g_2} = \tilde{\varepsilon }^{\mathcal {A}}_{-(g_1 + g_2)} \nonumber \\= & - \tilde{\varepsilon }^{\mathcal {A}}_{-g_1} = - \tilde{\varepsilon }^{\mathcal {A}}_{-g_2} = - \tilde{\varepsilon }^{\mathcal {A}}_{g_1 + g_2} = i \tilde{\varepsilon }^{\mathcal {A}}_1.\) 
